On actions and split extensions in varieties of hoops

BL-algebras were introduced by P. Hájek as the algebraic semantics of Basic Logic, the logic of continuous t-norms. They capture the common fragment of the three most relevant many-valued logics, namely Łukasiewicz Logic, Gödel Logic, and Product Logic. It is a classical result that, up to isomorphism, every continuous t-norm behaves locally as one of the following three fundamental continuous t-norms: the Łukasiewicz t-norm, defined by x *_L y = max{x + y - 1, 0}, the Gödel t-norm, defined by x *_G y = min{x, y}, and the product t-norm, defined by x *_P y = xy. Every continuous t-norm naturally induces a residuation, or implication, defined as follows: x -> y is the supremum of the set of all z in [0,1] such that z * x <= y. P. Hájek studied in detail the residuations associated with the three fundamental continuous t-norms, whose corresponding varieties of algebras may be described as follows: The variety MVAlg generated by ([0,1], *_L, ->_L, max, min, 0, 1) defines the class of MV-algebras, which form the algebraic semantics of Łukasiewicz Logic. Algebras in the variety GAlg generated by ([0,1], *_G, ->_G, max, min, 0, 1) are called Gödel algebras, and they form the equivalent algebraic semantics for Gödel Logic. The variety PAlg of product algebras is generated by ([0,1], *_P, ->_P, max, min, 0, 1) and its associated propositional calculus is Product Logic. Finally, P. Hájek introduced the variety BLAlg of BL-algebras, whose propositional calculus is Basic Logic. It was then shown that BLAlg is the variety generated by all algebras ([0,1], *, ->, max, min, 0, 1), where * is a continuous t-norm on [0,1]. From a categorical point of view, the variety BLAlg, as well as its subvarieties MVAlg, GAlg, and PAlg, determines an ideally exact category. Moreover, if 2 denotes the two-element Boolean algebra, then the semi-abelian (slice) categories (BLAlg/2), (MVAlg/2), (GAlg/2), and (PAlg/2) are equivalent, respectively, to the semi-abelian varieties of basic hoops, Wajsberg hoops, Godel hoops, and product hoops. A key notion that can be studied in the context of semi-abelian categories is that of internal action, which generalizes classical algebraic notions such as group or Lie algebra actions and provides a description of split extensions (and, therefore, of retractions) in algebraic terms, namely by means of semidirect products. The aim of this talk is to investigate actions and split extensions in the variety of hoops and in its subvarieties of basic, Wajsberg, Gödel, and product hoops. In particular, we focus on split extensions with strong section in the sense of W. Rump, which we describe in terms of strong external actions, i.e., pairs of maps satisfying a set of identities closely related to the axioms satisfied by the hoop. We prove that, for any hoop X, there is a natural isomorphism of functors between strong external actions on X and isomorphism classes of split extensions with strong section with kernel X. We also observe that the notion of split extension with strong section trivializes in the context of MV-algebras, whereas in the variety of Gödel hoops, strong external actions coincide with those in the variety of basic hoops. Eventually, we show a connection between the notion of strong external action in the variety of hoops and the semidirect product construction introduced by W. Rump in the category of L-algebras.